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nLab > Latest Changes: spectrum of an abelian category

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  1.  
    • CommentRowNumber1.
    • CommentAuthorzskoda
    • CommentTimeJul 8th 2010
    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 10th 2010

    Can the spectrum of an abelian category be seen as some collection of (prime) ideals in that category along the lines of what Awodey takes as the ideals of a category in the work mentioned at ideal completion?

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeJul 10th 2010
    • (edited Jul 10th 2010)

    I think no.

    • CommentRowNumber4.
    • CommentAuthorhilbertthm90
    • CommentTimeSep 11th 2012

    There is a question about these entries here.

    I thought I’d just open up the paper and answer their question, but I realized they’re right! These entries are fairly confusing on the point of removing the hypothesis of quasi-compactness. They all seem to say something like: Rosenberg proved you can reconstruct a scheme from its category of quasi-coherent sheaves in a more general situation by using a different spectrum.

    spectrum of an abelian category has 8 eight papers listed by Rosenberg and its not clear that any of them are the one that contain this theorem. Does anyone know which paper has it and/or what the exact hypotheses are?

    • CommentRowNumber5.
    • CommentAuthorzskoda
    • CommentTimeSep 11th 2012
    • (edited Sep 11th 2012)

    Look Prop. 8.2.1, p.32 in chapter II (page 65 of the file) in the book

    This is in quasicompact case. Usage of another spectrum goes somewhat beyond. I will give the reference for that later, now I have to take the bus in few minutes.

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeMay 9th 2013
    • CommentRowNumber7.
    • CommentAuthoradeelkh
    • CommentTimeOct 30th 2013

    I added the references

    • Martin Brandenburg, Rosenberg’s reconstruction theorem (after Gabber), 2013, arXiv.

    • John Calabrese, Michael Groechenig. Moduli problems in abelian categories and the reconstruction theorem. 2013. arXiv.

    to the page Gabriel-Rosenberg theorem.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 24th 2014

    I have expanded the Idea-section at spectrum of a triangulated category to provide a bit more of the basic background.

    Also I cross-linked with prime spectrum of a symmetric monoidal stable (infinity,1)-category

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeMay 24th 2014
    • (edited May 24th 2014)

    Well spectrum of a triangulated category should eventually not redirect to spectrum of a tensor triangulated category as these are different notions (and the nontensor variant is more basic, logically). In general, from a triangulated category you can reconstruct much less than from a tensor triangulated category as it is the case in abelian case as well.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 25th 2014
    • (edited May 25th 2014)

    The existing article pointed to Balmer and followup work, which is about the monoidal case, I added the explanation for that. The non-monoidal case still appears, as it did, I just added a brief piece of glue in an attempt to further clarify.

    But of course once we have some actual content in the entry, maybe we want to split it into two.

1 to 10 of 10

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